Lesson 5

Reasoning about Equations and Tape Diagrams (Part 2)

5.1: Algebra Talk: Seeing Structure (10 minutes)

Warm-up

This warm-up parallels the one in the previous lesson. The purpose of this Algebra Talk is to elicit strategies and understandings students have for solving equations. These understandings help students develop fluency and will be helpful later in this unit when students will need to be able to come up with ways to solve equations of this form. While four equations are given, it may not be possible to share every strategy. Consider gathering only two or three different strategies per problem, saving most of the time for the final question.

Students should understand the meaning of solution to an equation from grade 6 work as well as from work earlier in this unit, but this is a good opportunity to re-emphasize the idea.

In this string of equations, each equation has the same solution. Digging into why this is the case requires noticing and using the structure of the equations (MP7). Noticing and using the structure of an equation is an important part of fluency in solving equations.

Launch

Display one equation at a time. Give students 30 seconds of quiet think time for each equation and ask them to give a signal when they have an answer and a strategy. Keep all equations displayed throughout the talk. Follow with a whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Facing

Solve each equation mentally.

\(x -1 = 5 \)

\(2(x-1) = 10 \)

\(3(x-1) = 15 \)

\(500 = 100(x-1)\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

This discussion may go quickly, because students are likely to recognize similarities between this equation string and the one in the previous day’s warm-up.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

  • “Who can restate ___’s reasoning in a different way?”
  • “Did anyone have the same strategy but would explain it differently?”
  • “Did anyone solve the equation in a different way?”
  • “Does anyone want to add on to _____’s strategy?”
  • “Do you agree or disagree? Why?”
Speaking: MLR8 Discussion Supports.: Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . . ." Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)

5.2: More Situations and Diagrams (15 minutes)

Activity

The purpose of this activity is to work toward showing students that some situations can be represented by an equation of the form \(p(x+q)=r\) (or equivalent). In this activity, students are simply tasked with drawing a tape diagram to represent each situation. In the following activity, they will work with corresponding equations.

For each question, monitor for one student with a correct diagram. Press students to explain what any variables used to label the diagram represent in the situation.

Launch

Ensure students understand that the work of this task is to draw a tape diagram to represent each situation. There is no requirement to write an equation or solve a problem yet.

Arrange students in groups of 2. Give 5–10 minutes to work individually or with their partner, followed by a whole-class discussion.

Action and Expression: Develop Expression and Communication. Maintain a display of important terms, vocabulary, and examples. During the launch, take time to review examples of drawing a tape diagram based on situations from previous lessons that students will need to access for this activity. Consider providing step-by-step directions that generalize the process using student input and ideas.
Supports accessibility for: Memory; Language

Student Facing

Draw a tape diagram to represent each situation. For some of the situations, you need to decide what to represent with a variable.

  1. Each of 5 gift bags contains \(x\) pencils. Tyler adds 3 more pencils to each bag. Altogether, the gift bags contain 20 pencils.
  2. Noah drew an equilateral triangle with sides of length 5 inches. He wants to increase the length of each side by \(x\) inches so the triangle is still equilateral and has a perimeter of 20 inches.
  3. An art class charges each student \$3 to attend plus a fee for supplies. Today, \$20 was collected for the 5 students attending the class.
  4. Elena ran 20 miles this week, which was three times as far as Clare ran this week. Clare ran 5 more miles this week than she did last week. 

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

Ensure students understand that the work of this task is to draw a tape diagram to represent each situation. There is no requirement to write an equation or solve a problem yet.

Arrange students in groups of 2. Give 5–10 minutes to work individually or with their partner, followed by a whole-class discussion.

Action and Expression: Develop Expression and Communication. Maintain a display of important terms, vocabulary, and examples. During the launch, take time to review examples of drawing a tape diagram based on situations from previous lessons that students will need to access for this activity. Consider providing step-by-step directions that generalize the process using student input and ideas.
Supports accessibility for: Memory; Language

Student Facing

Draw a tape diagram to represent each situation. For some of the situations, you need to decide what to represent with a variable.

  1. Each of 5 gift bags contains \(x\) pencils. Tyler adds 3 more pencils to each bag. Altogether, the gift bags contain 20 pencils.
  2. Noah drew an equilateral triangle with sides of length 5 inches. He wants to increase the length of each side by \(x\) inches so the triangle is still equilateral and has a perimeter of 20 inches.
  3. An art class charges each student \$3 to attend plus a fee for supplies. Today, \$20 was collected for the 5 students attending the class.
  4. Elena ran 20 miles this week, which was three times as far as Clare ran this week. Clare ran 5 more miles this week than she did last week.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Select one student for each situation to present their correct diagram. Ensure that students explain the meaning of any variables used to label their diagram. Possible questions for discussion:

  • “For the situations with no \(x\), how did you decide what quantity to represent with variable?” (Think about which amount is unknown but has a relationship to one or more other amounts in the story.)
  • “What does the variable you used to label the diagram represent in the story?”
  • “Did any situations have the same diagrams? How can you tell from the story that the diagrams would be the same?” (Same number of equal parts, same amount for the total.)
Speaking, Representing, Reading: MLR1 Stronger and Clearer Each Time. Ask students to explain to a partner how they created the tape diagram to represent the situation “An art class charges each student \$3 to attend plus a fee for supplies. Today, \$20 was collected for the 5 students attending the class.” Ask listeners to press for details in the arrangement of the grouped quantities (e.g., “Explain how you chose what values go in each box.”). When roles are switched, listeners can press for details in what “\(x\)” represents in the diagram. Allow students to revise their diagrams, if necessary, based on the feedback they received from their partner. Once their revision is complete, invite students to turn to a new partner to explain their revised diagram. This will help students productively engage in discussion as they make connections between written situations and visual diagrams.
Design Principle(s): Optimize output (for explanation); Cultivate conversation

5.3: More Situations, Diagrams, and Equations (10 minutes)

Activity

This activity is a continuation of the previous one. Students match each situation from the previous activity with an equation, solve the equation by any method that makes sense to them, and interpret the meaning of the solution. Students are still using any method that makes sense to them to reason about a solution. In later lessons, a hanger diagram representation will be used to justify more efficient methods for solving. 

For each equation, monitor for a student using their diagram to reason about the solution and a student using the structure of the equation to reason about the solution.

Launch

Keep students in the same groups. 5 minutes to work individually or with a partner, followed by a whole-class discussion.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as “To find the solution, first, I _____ because...”, “I made this match because I noticed...”, “Why did you...?”, or “I agree/disagree because…”
Supports accessibility for: Language; Social-emotional skills
Speaking, Representing: MLR2 Collect and Display. As students share their ideas about how the equations match the situations, listen for and collect students’ description of the situation (e.g., “5 gift bags, \(x\) pencils, adds 3 more, 20 pencils”) with the corresponding equation. Remind students to borrow language from the displayed examples while describing what each solution tells about the situation, after the matching is complete. This will help students make connection between language, diagrams, and equations.
Design Principle(s): Support sense-making; Maximize meta-awareness

Student Facing

Each situation in the previous activity is represented by one of the equations.

  • \((x+3) \boldcdot 5 = 20\)
  • \(3(x+5)=20\)
  1. Match each situation to an equation.
  2. Find the solution to each equation. Use your diagrams to help you reason.
  3. What does each solution tell you about its situation?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Han, his sister, his dad, and his grandmother step onto a crowded bus with only 3 open seats for a 42-minute ride. They decide Han’s grandmother should sit for the entire ride. Han, his sister, and his dad take turns sitting in the remaining two seats, and Han’s dad sits 1.5 times as long as both Han and his sister. How many minutes did each one spend sitting?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

For each equation, ask one student who reasoned with the diagram and one who reasoned only about the equation to explain their solutions. Display the diagram and the equation side by side, drawing connections between the two representations. If no students bring up one or both of these approaches, demonstrate the maneuvers on a diagram side by side with the maneuvers on the corresponding equation. For example, “I divided the number of gift bags by 5, leaving me with 4 pencils per gift bag. Since Tyler added 3 pencils to each gift bag, there must have been 1 pencil in each gift bag to start,” can be shown on a tape diagram and on a corresponding equation. It is not necessary to invoke the more abstract language of “doing the same thing to each side” of an equation yet.

Lesson Synthesis

Lesson Synthesis

Display one of the situations from the lesson and its corresponding equation. Ask students to explain:

  • “What does each number and letter in the equation represent in the situation?”
  • “What is the reason for each operation (multiplication or addition) used in the equation?”
  • “What is the solution to the equation? What does it mean to be a solution to an equation? What does the solution represent in the situation?”

5.4: Cool-down - More Finding Solutions (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Equations with parentheses can represent a variety of situations. 

  1. Lin volunteers at a hospital and is preparing toy baskets for children who are patients. She adds 2 items to each basket, after which the supervisor’s list shows that 140 toys have been packed into a group of 10 baskets. Lin wants to know how many toys were in each basket before she added the items.
  2. A large store has the same number of workers on each of 2 teams to handle different shifts. They decide to add 10 workers to each team, bringing the total number of workers to 140. An executive at the company that runs this chain of stores wants to know how many employees were in each team before the increase.

Each bag in the first story has an unknown number of toys, \(x\), that is increased by 2. Then ten groups of \(x+2\) give a total of 140 toys. An equation representing this situation is \(10(x+2)=140\). Since 10 times a number is 140, that number is 14, which is the total number of items in each bag. Before Lin added the 2 items there were \(14 - 2\) or 12 toys in each bag.

The executive in the second story knows that the size of each team of \(y\) employees has been increased by 10. There are now 2 teams of \(y+10\) each. An equation representing this situation is \(2(y+10)=140\). Since 2 times an amount is 140, that amount is 70, which is the new size of each team. The value of \(y\) is \(70-10\) or 60. There were 60 employees on each team before the increase.